Colloquium: Phononics: Manipulating heat flow with electronic analogs and beyond

The form of energy termed heat that typically derives from lattice vibrations, i.e., phonons, is usually considered as waste energy and, moreover, deleterious to information processing. However, in this Colloquium, an attempt is made to rebut this common view: By use of tailored models it is demonstrated that phonons can be manipulated similarly to electrons and photons, thus enabling controlled heat transport. Moreover, it is explained that phonons can be put to beneficial use to carry and process information. In the first part ways are presented to control heat transport and to process information for physical systems which are driven by a temperature bias. In particular, a toolkit of familiar electronic analogs for use of phononics is put forward, i.e., phononic devices are described which act as thermal diodes, thermal transistors, thermal logic gates, and thermal memories. These concepts are then put to work to transport, control, and rectify heat in physically realistic nanosystems by devising practical designs of hybrid nanostructures that permit the operation of functional phononic devices; the first experimental realizations are also reported. Next, richer possibilities to manipulate heat flow by use of time-varying thermal bath temperatures or various other external fields are discussed. These give rise to many intriguing phononic nonequilibrium phenomena such as, for example, the directed shuttling of heat, geometrical phase-induced heat pumping, or the phonon Hall effect, which may all find their way into operation with electronic analogs.

Figure 1

Sketch of the modus operandi of a thermal diode. When the left end of the diode is at a higher temperature as compared to its right counterpart, heat is allowed to flow almost freely. In contrast, when the right end is made hotter than the left,the transduction of heat becomes strongly diminished.

Figure 2

Concept and experimental setup of a nanoscale thermal diode. An asymmetric nanotube is placed over two electrodes with one serving as a heater. For further details on the experimental procedures and the experimental findings see the experiment by 16 and the detailed discussion in Sec. 3. Adapted from 16.

Figure 3

Concept of a thermal diode. (a) Blueprint of an efficient two-segment thermal diode composed of two different Frenkel-Kontorova chains. The left segment consisting of a chain of particles subjected to a strong cosinusoidal varying on-site potential (illustrated by the large wavy curve) is connected to the right segment possessing a relatively weak on-site potential. (b) In the case in which the temperature $T_L$ in the left segment is colder than the corresponding right temperature $T_R$, i.e., $T_L<T_R$, the power spectrum of the particle motions of the left segment is weighted at high frequencies. This occurs because of the difficulty experienced by the particles there in overcoming the large barriers of the on-site potential. In contrast, the power spectrum of the right segment is weighted at low frequencies. As a result, the overlap of the spectra is weak, implying that the heat current $J$ becomes strongly diminished. (c) Here the situation is opposite to that in (b). With $T_L>T_R$ the particles can now move almost freely between neighboring barriers. Consequently the power spectrum extends to much lower frequencies, yielding an appreciable overlap with the right segment. This in turn causes a sizable heat current. (d) Heat current $J$ vs the relative temperature bias $\Delta$, as defined in the inset, for three different values of the reference temperature $T_0$. Adapted from 68 and 117.

Figure 4

Temperature profiles within a FK-FK thermal diode. The temperature profiles for various interface elastic constants $k_{\mathrm{int}}=0.01$ (squares), 0.05 (circles), and 0.2 (triangles). $T_0=0.07$ and $N=100$. The open symbols correspond to a positive temperature bias $\Delta =0.5$, while the solid symbols correspond to a negative temperature bias $\Delta =-0.5$. The dashed line indicates the interface. The temperature jumps at the interface are due to the Kapitza resistance which is addressed in Sec. 2a2. Adapted from 68.

Figure 5

Concept of a thermal transistor. (a) A one-dimensional lattice is coupled at its two ends to heat baths at temperatures $T_D$ and $T_S$ with $T_D>T_S$. A third heat bath with temperature $T_S<T_O<T_D$ can be used to control the temperature at the node $O$, so as to control the heat currents $J_D$ and $J_S$. (b) In an extended regime, as the temperature $T_O$ increases, the thermal bias ($T_D-T_O$) at the interface decreases while the power spectrum in the left segment of the control node increasingly matches with the power spectrum of the neighboring segment that is connected to the drain $D$. This behavior is depicted in the insets where the corresponding power spectra of the left-side and right-side interface particles are depicted for three representative values of $T_O$, as shown by the three arrows. The resulting drain current $J_D$ increases with increasing overlap until it reaches a maximum and starts to decrease again.

Figure 6

Thermal transistor. (a) Sketch of a thermal transistor device. Just as in the case of an electronic transistor, it consists of two segments (the source and the drain) and, as well, a third segment (the gate) where the control signal is injected. The temperatures $T_D$ and $T_S$ are fixed at high, $T_{+}$, and low, $T_{-}$, values. Negative differential thermal resistance (NDTR) occurs at the interface between $O$ and $O^{\prime }$. This then makes it possible that, over a wide regime of parameters, when the gate temperature $T_G$ rises, not only $J_S$ but also $J_D$ may increase. Use of different system parameters then allows different functions, this being either a thermal switch (b) or a thermal modulator (c). (b) Function of a thermal switch: At three working points where $T_G=0.04$, 0.09, and 0.14, the heat current $J_D$ equals $J_S$, so that $J_G=J_S-J_D$ vanishes identically. These three working points correspond to (stable) “off,” (unstable) “semi-on,” and (stable) “on” states, at which $J_D$ differs substantially. We can switch, i.e., forbid or allow heat flowing by setting $T_G$ at these different values. (c) Function of a thermal modulator. Over a wide temperature interval of gate temperature $T_G$, depicted via the hatched working region, the heat current $J_G$ remains very small, i.e., it remains inside the hatched regime, while the two heat currents $J_S$ and $J_D$ can be continuously controlled from low to high values. Adapted from 69.

Figure 7

Negative response and thermal NOT gate. (a) Temperature of the node $O^{\prime }$ as a function of $T_G$ for the setup shown in Fig. 6a. In a wide range of $T_G$, $T_{O^{\prime }}$ decreases as $T_G$ increases. (b) Function of the thermal NOT gate. The thin line indicates the function of an ideal NOT gate. Inset: Structure of a two-resistor voltage divider, the counterpart of a temperature divider, which supplies a voltage lower than that of the battery. The output of the voltage divider is $V_{\mathrm{out}}=V{R}_2/(R_1+R_2)$. Adapted from 116.

Figure 8

Write-and-read cycle in a thermal memory. (a) Illustration of the four stages of a complete writing-reading process. The left and right ends of this thermal memory are coupled to heat baths at fixed temperatures $T=0.03$ and 0.3, respectively. During the initializing stage the system relaxes toward its stationary state with the temperature at site $O$ being $T_O=0.18$. The different stages during this cycle are indicated by dashed vertical lines in (b) and (c). (b) Time evolution of the local temperature $T_O(t)$ (dotted wine color) of the central particle at site $O$, the temperature of the writer $T_{\mathrm{writer}}(t)$ (solid), and the temperature of the reader $T_{\mathrm{reader}}(t)$ (dashed) during a complete write-and-read process. The time-dependent local temperature of the particle (with unit mass) at site $O$ is obtained by integrating and averaging the squared velocity $v_O^2(t)$ over a short time span (${10}^4$ time units), centered at time instant $t$. The writer and reader are realized with short lattice segments, while $T_{\mathrm{writer}}$ and $T_{\mathrm{reader}}$ are the averaged temperatures of all particles within the writer and reader, respectively. During the writing stage the node $O$ is coupled to one end of the writer, whose opposite end is coupled to a heat bath set at the off value $T_{\mathrm{off}}=0.04$. During the reading stage, the reader set at initial temperature 0.11 is attached to the node $O$. (c) The opposite write-and-read cycle with the writer set at the on value with $T_{\mathrm{on}}=0.27$. Adapted from 118.

Figure 9

Schematic system setups for thermal diodes devised from asymmetric nanostructures: (a) a typical $(n,0)/(2n,0)$ carbon nanotube junction; (b) a carbon-nanocone-based thermal rectifier; (c) two asymmetric graphene nanoribbons (GNRs), a trapezium-shaped GNR, and, as well, a structure made of two rectangular GNRs of different widths; (d) a topological configuration based on a Möbius graphene stripe. The parts ($T_L$ and $T_R$) denote the corresponding heat bath regions. The bath temperatures of the two ends are denoted as $T_T$ (top) and $T_B$ (bottom) in (b) and (c), and as $T_L$ (left) and $T_R$ (right) in (a) and (d), respectively. Adapted from 121, 128, 129, and 53.

Figure 10

Experimental realization of thermal rectification in nanotube junctions. (a) Scanning-electron-microscope images of boron nitride nanotubes (BNNTs) after deposition of ${\mathrm{C}}_9{\mathrm{H}}_{16}\mathrm{Pt}$. (b) Graphical representation of the temperature changes of the heater ($\Delta T_h$) and sensor ($\Delta T_s$) for the nanotubes before and after deposition of ${\mathrm{C}}_9{\mathrm{H}}_{16}\mathrm{Pt}$. Adapted from 16.

Figure 11

Experimental realization of the thermal memory. (a) Scanning-electron-microscope image of the memory device, consisting of a ${\mathrm{VO}}_2$ nanobeam connecting the input terminal (left side) and output terminal (right side). An equivalent thermal circuit is also depicted in the image. (b) Output temperature $T_{\mathrm{out}}$ as a function of input temperature $T_{\mathrm{in}}$, upon heating (solid symbols) and cooling (open symbols). The characteristics of the thermal memory are tuned under three different voltage biases, as indicated. (c) Write-and-readout process of logical 1 ($=\mathrm{\text{high-temperature state}}$) and 0 ($=\mathrm{\text{low-temperature state}}$). Upper part, $T_{\mathrm{in}}(t)$ for writing: To write a high- ($=1$) temperature state to the output terminal, a heating pulse generated by increasing the heating current was applied to the input terminal to set $T_{\mathrm{in}}$ to 435 K. To switch the output terminal to a low-temperature state, a cooling pulse was applied to the input terminal to set $T_{\mathrm{in}}$ to 300 K by natural cooling of $T_{\mathrm{in}}$ to the substrate temperature $T_{\mathrm{base}}$. Lower part, $T_{\mathrm{out}}(t)$ for reading: When $T_{\mathrm{in}}$ was returned to 360 K for reading, the high- (low-) temperature of $\sim 332.1\pm 0.1\mathrm{K}$ ($\sim 317\pm 0.8\mathrm{K}$) was retained and could be read out at the output terminal $T_{\mathrm{out}}$. Here the voltage bias is 0.04 V. Adapted from 126.

Figure 12

Action of a heat Brownian motor in two coupled asymmetric FK-FK lattices. The heat baths are subjected to periodic modulations in the forms $T_L(t)=T_0[1+\Delta +A\mathrm{sgn}(\sin \omega t)]$ and $T_R(t)=T_0(1-\Delta )$. The dimensionless reference temperature is set as $T_0=0.09$ (see Appendix  for the expressions in corresponding dimensionless units). Note that in distinct contrast to Fig. 3d the ratchet effect now yields with a modulation strength $A\ne 0$ a nonvanishing heat flow at zero bias $\Delta =0$. Applying a substantial rocking strength $A=0.5$, the current bias characteristics can be manipulated to inhibit a negative differential thermal resistance (NDTR) regime at negative bias values $\Delta$. Adapted from 70.

Figure 13

Quantum heat shuttling. (a) Setup of a molecular junction acting as a heat shuttle. The short molecular wire is composed of a single electronic level $E_1$ only that can be gated and a single phonon mode at the fixed vibrational frequency ${\omega }_1=1.4\times {10}^{14}{\mathrm{s}}^{-1}$, typical for a carbon-carbon bond. The lead temperatures $T_{L(R)}(t)$ are subjected to time-periodic modulations. (b) Quantum thermal ratchet effect for this molecular junction. Total directed heat current $J_Q$ as a function of static thermal bias $\Delta T_{\mathrm{bias}}$ for different driving amplitude strengths $A$. The heat baths are modulated as $T_L(t)=T_0+\frac{1}{2}\Delta T_{\mathrm{bias}}+A\cos \Omega t$ and $T_R(t)=T_0-\frac{1}{2}\Delta T_{\mathrm{bias}}$. The reference temperature is set as $T_0=300\mathrm{K}$ and the electronic wire level is set as $E_1-\mu =0.138\mathrm{eV}$, where $\mu$ is the chemical potential. $J_Q$ is independent of the driving frequency $\Omega$ as a consequence of ballistic heat transfer in the adiabatic limit. From 133.

Figure 14

Quantum heat pumping via a geometrical phase. (a) A schematic representation of a single molecular junction. (b) Quantum heat transfer across the molecular junction is generated via an adiabatic two-parameter variation of the left bath temperature $T_L(t)$ and the right bath temperature $T_R(t)$, which maps onto a closed circle. The arrow indicates the direction of the modulation protocol. (c) Geometrical-phase-induced heat current $J_{\mathrm{geom}}$ vs the angular modulation frequency $\Omega$. The straight line corresponds to the analytic result while the open circles denote the simulation results. For further details see 90.

Figure 15

A schematic illustration of the phonon Hall effect with heat flowing along the direction indicated by the large arrow.

Figure 16

Temperature-dependent power spectra. The variation of the power spectrum at different temperatures vs the angular frequency $\omega$ (both in dimensionless units) in a FK lattice with 10 000 sites for two different sets of parameters in (a) and (b). The features of these nonlinear FK power spectra provide the seed for the modus operandi in a thermal diode setup as discussed in Sec 2a1. (a) FK coupling strength of $V=5$ and a strength for the spring constant of $k=1$; (b) coupling strength $V=1$ and a spring constant value set at $k=0.2$.